Bass Vs speaker efficiency

For drivers, no, but in speaker design you can often trade sensitivity for deeper bass. Less sensitive = deeper bass.

A great example of this is the Magico S1 Mk II, which is pretty inefficient (86 dB) but achieves remarkable bass for the size.  (sits back and waits for the Magico shills to go absolutely bonkers).

No relationship at all. 

Not many things in audio depend on just one other thing. Being connected, most things depend on being connected to something. But there must be at least a half a dozen things I have done that have made the bass tighter and more articulate but had nothing to do with efficiency. So the answer is no.

“Tight” bass no. But efficiency and box size have a relationship for deep bass. The pressure increases faster in a smaller box and the driver has to work harder to overcome it. 

'Tight' bass is a coloration and has nothing to do with efficiency.